Integrals of monomials over the orthogonal group

TL;DR

This paper introduces a recursion formula for evaluating invariant integrals of monomials over the orthogonal group O(N), expanding the tools available for such calculations and expressing results as finite sums of partial fractions.

Contribution

A new recursion formula for integrals over O(N) that generalizes existing methods and simplifies the evaluation of monomial integrals.

Findings

Recursion formula enables evaluation of integrals as finite sums of partial fractions.

Extends existing integration formulas for the orthogonal group.

Provides a systematic method for invariant monomial integrals.

Abstract

A recursion formula is derived which allows to evaluate invariant integrals over the orthogonal group O(N), where the integrand is an arbitrary finite monomial in the matrix elements of the group. The value of such an integral is expressible as a finite sum of partial fractions in $N$. The recursion formula largely extends presently available integration formulas for the orthogonal group.